Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $\big (\|Y\|, Y/ \|Y\| )^\top $ and its best approximation by a distribution of a vector $\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.

翻译：大多数关于检验$p$维随机向量$Y$分布中球对称性假设的研究，都集中于针对精确球对称性零假设的统计检验。本文采用不同视角，提出了一种基于向量$\big (\|Y\|, Y/ \|Y\| )^\top $分布与球对称随机向量$Y_s$对应的向量$\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $分布的最佳逼近之间最小距离的偏离度量方法。我们建立了具有相应统计保证（由渐近理论提供）的最小距离估计量，并通过模拟研究和实际数据案例证明了该方法的适用性。